microeconomics, economics homework help

Question 1
A production function for pianos is given by: Q = 10K0.5L0.5.
a. If the firm is using 1 unit of capital and 16 units of labour, how many pianos are produced?
b. If 4 units of capital are used, how many units of labour are needed to match production?
Question 2
A paper company needs to produce 2,000 reams of paper for its customers. The firm’s long-run production function and the marginal productivities of the inputs are:
Q = 4K0.75L0.25 MPL = K0.75/L0.75 MPK = 3L0.25/K0.25
where Q is the number of reams produced, K is the quantity of capital rented and L is the quantity of labour hired. The weekly cost function is C = 20K + 2L, where C is the total cost of production.
a. What ratio of capital to labour minimizes total costs?
b. How much labour and capital will need to be employed to produce 2,000 reams a week?
Question 3
Suppose a firm has a ‘Cobb-Douglas’ production function, and marginal productivities, as
follows: Q = 0.5K0.5L0.5 MPL = 0.25K0.5/L0.5 MPK = 0.25L0.5/K0.5
Labour costs W = $10 per hour and capital costs R = $40 per hour. The production target for the firm is 100 units of output.
a. If the firm is operating efficiently (i.e. minimizes total costs), how many units of labour will be employed? How many units of capital? What is the cost of producing 100 units of output?
b. What will happen to the firm’s production mix and total cost if the rental rate suddenly drops to $10 per unit of capital (if it still wishes to produce 100 units of output)?
c. Does production exhibit constant, decreasing or increasing returns to scale?